(Measurement and Measured Value)
Measurement is to compare an amount to be measured (hereinafter “measurand”) of an object or a phenomenon to be measured (hereinafter “measuring object”) with an amount used as a reference to express with a numerical value or a reference sign, and a value obtained by measurement is called “measured value”.
(Measuring System)
A system which is composed with objects and necessary functions while collecting the functions for the purpose of measurement of a certain amount is called “measuring system”. In FIG. 1, a schematic view of the measuring system is shown. The measuring system includes all of the measurer, measuring instrument, measuring object, and measurement environment.
(Error)
When measurement is performed, an error is included in a measured value. An error is a difference between the measured value and a true value. Here, the true value is an ideal or virtual value introduced for convenience, and actually, the error should be evaluated in a state where the true value is unknown.
(Error Factor)
Error factors in each constituent element of the measuring system are shown in Table 1. Error occurs with these error factors.
TABLE 1ConstituentElements ofMeasuring systemError FactorsMeasuringIncompleteness of measurement principleInstrumentIncompleteness of constitution or operationof measuring instrumentChange of measuring instrument according to dif-ference or variation of measurement environment(s)Measuring ObjectChange of measuring object itself according to dif-ference or variation of measurement environment(s)MeasurerHabit or measurement mistake of measurer
(Classification of Errors)
In general, errors can be dealt with after dividing into systematic errors and random errors.
(Systematic Error)
Systematic errors are a general term of errors arising from factors which bias a population mean of measured values from the true value, out of various error factors. Errors of a measuring instrument, personal error, and the like are belong to this type.
(Random Error)
Random errors are errors which occur from causes which cannot be found out and random errors emerge as dispersion of measured values. Because of this type of errors, even when measurement is repeated in the same conditions, respective measured values are uneven. Since the random errors emerge from causes which cannot be found out and extremely various causes, normally, they are dealt with probabilistically and statistically.
(Model of Measured Value and Error)
Now, with respect to a given measurand of given measuring objects, suppose that the true value is ν, measured values are x, errors in the measured values are ε, and systematic errors of ε are εS and random errors εR. A distribution of the random errors εR is supposed to be the normal distribution, an average of the measured values x is expressed as xaν, and a standard deviation is expressed as σ. In this case, a measured value x and an error ε can be modeled as shown in FIG. 2. Here, the error ε is the sum of a systematic error εS and a random error εR, that is, it is expressed in Equation 1, and the systematic error εS is a difference between the population mean xav and the true value ν, that is, it is expressed in Equation 2.ε=εS+εR  (Equation 1)εS=xaνi −V  (Equation 2)
(Removal of Systematic Errors)
Since the systematic errors are errors which occur according to given regular relationships and have reproducibility, there is a possibility that the measured value can be corrected to a value close to the true value by evaluating an influence of the cause. Therefore, efforts to remove the systematic errors have been performed by utilizing estimation from measurement conditions or measurement theory, or actual measurements where measurement conditions, apparatus, method, or the like are changed.
(Removal of Random Errors)
Since the random errors are errors which occur randomly or probabilistically because of unspecified large number of causes and disperse every measurement, it is impossible to remove them after measurement. However, they can be reduced relatively easily. Because in many cases, the distribution of the random errors may be regarded as the normal distribution and they are considered to disperse at probabilities of the same degree in positive and negative directions, if the average of the results is taken by performing the same measurement many times, it is considered the errors cancel each other to become smaller.
(Correction and Corrected Value)
To compensate a systematic error, a value algebraically added to a measured value or adding a value to it is called correction. Hereinafter, removing a systematic error in a measured value is referred to as “correction”, and a corrected measured value is referred to as “corrected value”.
(Error in Measured Value and Transformation of Measured Value)
If errors in measured values can not be processed appropriately, it is not possible to ensure validity in transformation results of the measured values. In the following description, methods of dealing with errors in measured values in conventional methodology will be described and their problems will be discussed.
(Correction by Calibration)
Calibration is a process to obtain a relationship between a value indicated by a measuring instrument and a value indicated by a measurement standard or standard sample (hereinafter, “standard value”), and it can be positioned as a means for removing the systematic error derived from a measuring instrument. Also, the relation connecting the standard values and the measured values, which is obtained by calibration, is called a calibration curve. In FIG. 3, an example of the calibration curve is shown. When the standard values indicated by the measurement standard or the like are si and the measured values of the measuring instrument to be calibrated are yi, a calibration curve 11 is obtained by plotting points (si, yi) on a standard value-measured value space and fitting a curve to them. In FIG. 3, a straight line 12 indicating y=s is also shown for reference.
(Limit of Correction by Calibration)
In the process of calibration, since it is assumed that there are a measurement standard and a standard material, correction can not be performed in a condition where they are not available. Also, if the measurement environment has changed, not only the measurement instrument but also the measuring object are affected by the change, however, calibration is only a means for removing the systematic error derived from the measuring instrument. As such, calibration is the process for transforming a measured value in a measuring system where measuring is actually performed (hereinafter, “actual measuring system”) into a measured value in a measuring system where calibration is performed by using the calibration curve (hereinafter, “calibration system”), and therefore it is not the means for correcting a measured value.
(Correction by Analytical Method)
One of the means for correcting a measured value is an analytical method. For example, a data analysis method or the like based on the design of experiments corresponds to this approach. In FIG. 4, a schematic view of correction based on the analytical method is shown. In the analytical method, first, change of an actually measured value is considered as effects of a finite number of error factors α1, α2 . . . αm, and a math model composed of a linear combination of the products of error factors and coefficients indicating degree of the effects is made. And coefficients composing the math model are determined by data analysis of the actually measured values, and the relationship between true values (criterion variables) ν1, ν2 . . . νn and measured values (explanatory variables) x1, x2 . . . xn are clearly obtained.
(Limit of Correction by Analytical Method)
The analytical method is a way of thinking adopting the symbolism where “If the conditions can be completely transformed into symbols and rules controlling the conditions can be completely clarified, it is possible to explain all phenomena completely”. However, in the analytical method, there are problems such as (i) there is no basis for selecting a finite number of error factors from an unspecified number of error factors, (ii) since independence of error factors is unclear, the validity of the math model is not ensured, (iii) since there are many uncertain elements in experiment and observation, convergence of solution is not ensured, or the like. Correction based on the analytical method is extremely complicated and always has uncertainty associated with symbol processing, and therefore it must be a method where the reliability of result of the process is poor.
(Summary of Conventional Correction Methods of Measured Values)
As discussed above, the correction methods of measured values in the present metrology are only limited methods, such as “method where a special instrument or the like is required”, “method where the systematic error derived from the measuring instrument is a subject”, “method which works only in a special measuring system”, “method where the reliability of result of correction can not be ensured”, or the like. The limited correction method of measured values means measured values are kept in the measuring system as a specific existence. In the following description, “lack of consistency of a measured value” expresses that a measured value is an existence limited in a specific measuring system and it is not a universal existence over measuring systems. When a measured value in a different measuring system is transformed without consistency of a measured value, the validity in the transformation result can not be ensured. In the following description, such a condition in transformation of a measured value is expressed as “lack of validity in a transformation result of a measured value”.